Jump To Content
AIEEE Prep

Community

AIEEE Prep

Primary Navigation

Perfect Squares

There has been a huge surge in the number of questions about perfect squares, in almost all the exams. The basic trick to any such question is assuming the number as a perfect square of an integer k and then using techniques of completion of square and then the formula of (a^2-b^2) and solving using divisibility theory

Example I: Find all natural n such that n(n+16) is a perfect square

Step 1: n(n+16)=k^2

Step 2: (n^2+2.8.n+8^2)-8^2=k^2

Step 3: (n+8+k)(n+8-k)=64

see now lhs and rhs both are integers then both of (n+8-k) and (n+8+k) are divisors of 64. But note that we add the two equations we will get 2n+16, so the sum of two divisors should be even hence both divisors even or both odd

so n+8+k=32,16,8,4,2 and n+8-k=2,4,8,16,32

but see this n is positive hence k is positive, thus n+8+k>n+8-k
so only two options

and solving we get 2n+16=34,20
so n=9,2

Note : The source of this problem is Pomona Wisconsin mathematics talent search exam!


Practice problem!!
Find the sum of all such positive integers m’s such that m^2+25m+19 is a perfect square

Now we will extend the method to other kinds of problems
Basically what we used in the above problem is difference of square method

lets take an example
x^6=y^2+127, find the no of pairs of positive integers (x,y)

first step in this problem is recognizing that 127 is a prime
then we move to
(x^3+y)(x^3-y)=127
so clearly 2x^3=128
x = 4 and y = 63
so one pair (4,63)

  • Your comment will be modifiable for 10 minutes after posted.

Page Author

Avatar
Sureshbala
Name
Sureshbala

From Here You Can…

Information

Most Recent Related Content

Published In…

This work is public domain.